Positive-density conjecture for Hodge–Witt reduction

Let $K$ be a number field and let $X/K$ be a smooth projective variety. Say that $X$ has Hodge–Witt reduction at a prime of $K$ when its smooth reduction at that prime is Hodge–Witt. Hodge–Witt reduction conjecture. $X$ has Hodge–Witt reduction modulo a set of primes of $K$ of positive density. Since ordinary varieties are Hodge–Witt, this is presented as a weaker version of Serre's conjecture on positive-density ordinary reduction. The paper formulates the statement as a conjectural weakening; no resolution is given in the supplied text.